The proposition REF  shows that the jump matrix \(V^{(sp)}\)  uniformly goes to \(I\)  outside the neighborhood of \(\pm z_0\) ,
then following the result in [1]}, the above RHP10 is solvable. The main contribution to the \(M^{sp}(z;\eta )\)  comes from
a local RH problem near \(\pm z_0\) , see Figure REF . We simply describe the process of construction for the solution of the RHP10,
see [1]} for the detail. We decompose the
jump matrix \( V^{(sp)}=(b_-)^{-1}b_+\) , and set
\(w_\pm =\pm (b_\pm -I), \ \ \ w=w_++w_-,\) 
